More generally, one can ask whether the converse of the Gitik-Mitra-Rips-Sageev Theorem is true. A subgroup H has width n if there exists a collection of n (essentially distinct) conjugates of H such that the intersection of any two elements of the collection is infinite and n is the maximum possible such integer. The Gitik-Mitra-Rips-Sageev Theorem asserts that quasiconvex subgroups of word-hyperbolic groups have finite width. So the stronger question asks whether quasiconvexity is equivalent to finite width in word-hyperbolic groups. See Gadde Swarup’s comment below.

2-Manifolds

Can the action of the mapping class group of a surface on the unit tangent bundle of the surface be realized by diffeomorphisms? See Ian Agol’s comment on 7/3/08.

Which (infinite, non-cyclic) subgroups of the mapping class group of a surface can be realized by subgroups of the diffeomorphism or self-homeomorphism group? See 7/3/08.

Does Hempel’s algorithm for constructing paths in the curve complex produce geodesics? See 1/16/08.

3-Manifolds

Is the subgroup of a mapping class group generated by the mapping class groups of two handlebodies an amalgamated free product of the two groups? (Originally due to Yair Minsky) See 7/14/08.

Given the commensurator of (the exteriors of) two knots in S^3, how do the fillings obtained by the lifts of their meridians compare? In particular, what happens for Aitchison and Rubinstein’s dodecahedral knots? See 6/30/08.

Is every minimal layered triangulation of a lens space minimal among all triangulations? See 5/21/08.

Is every mapping class group of a 3-manifold isomorphic to a subgroup of a surface mapping class group? See 2/15/08.

Does every cusped finite-volume hyperbolic three-manifold admit a geometric triangulation? (Here the ideal tetrahedra must have positive volume.) Suggested by Saul Schleimer.

Does every hyperbolic 3-manifold have a finite cover that is Haken/fibered/positive first Betti number? See 11/19/07. Seems to have a positive answer by Agol’s work?

List all closed hyperbolic 3-manifolds with volume less than … See 11/18/07.

Give a conceptual explanation for the normalization factor in the version of the framed Kontsevich invariant which gets fed into the LMO/Aarhus construction. Is it the unique normalization which would give rise to a 3-manifold invariant? See 11/09/10.

3-Manifold Heegaard splittings

Given a Heegaard splitting of a hyperbolic 3-manifold (i.e. if we can choose a nice metric onM), how “nice” is the map from the marked placement space to Teichmuller space? See 12/20/08.

Find an irreducible Heegaard splitting that does not have a Heegaard diagram satisfying the weak rectangle condition. See 12/5/08.

Given a Heegaard surface, characterize the set of loops in its curve complex that are homotopy trivial in the ambient manifold. (Originally due to Yair Minsky) Characterize the loops that are isotopic to geodesics. See 7/14/08.

When is the non-minimal Heegaard surface produced by lifting a minimal bridge surface to the double branched cover reducible? See 7/9/08.

Is there a connection between the existence of a tight/fillable/etc. contact structure and the existence of a high distance Heegaard splitting? See 5/20/08.

Does every 3-manifold have a Heegaard splitting such that the set of primitive disks has maximal dimension in the curve complex? See 4/29/08.

Prove something about Heegaard splittings using splitting homomorphisms. See 3/13/08.

Is there a hyperbolic 3-manifold with a Heegaard splitting such that each handlebody induces a redundant set of generators for the fundamental group? See 3/3/08.

If, given two non-isotopic Heegaard splittings of a fixed 3-manifold, we take the connect sum of each with a fixed Heegaard splitting of a second 3-manifold, can the resulting Heegaard splittings be isotopic? See 3/3/08.

Does the existence of a tight contact structure imply the existence of a strongly irreducible Heegaard splitting or vice versa? See 2/21/08.

Classify the set of all open book decompositions that induce Heegaard splittings in a fixed isotopy class. See 2/21/08.

Does the short exact sequence associated with the mapping class group of a Heegaard splitting always split? See 2/15/08.

Is there a hyperbolic 3-manifold whose rank is strictly less than its genus? (And if so, how big can the difference be?) See 2/11/08.

Given an irreducible Heegaard splitting of a 3-manifold with a torus boundary such that the Heegaard splitting is stabilized for infinitely many Dehn fillings, is the Heegaard splitting necessarily PADed? See 2/5/08.

Is there a direct proof (i.e. not using geometrization) that 3-manifolds with Heegaard splittings of distance three or greater are hyperbolic? See 11/29/07.

Is there a 3-manifold with an irreducible, weakly reducible Heegaard splitting of non-minimal genus? See 11/19/07.

Is there an algorithm to determine the Heegaard genus of a 3-manifold? See 11/19/07.

Can the mapping class group of an irreducible Heegaard splitting act trivially on the fundamental groups of the handlebodies of the splitting? See 11/19/07.

3-Manifold and knot/link concordance

Given a 3-manifold that can be embedded in R^4, can it be embedded so that the restriction of the height function is a Morse function inducing a minimal genus Heegaard splitting? See 6/13/08.

Given a smoothly embedded 3-sphere in the 4-sphere, does it bound a smoothly-embedded 4-ball? This is the Schoenflies problem in dimension four. See 11/05/11.

Which homology 3-sphere bound homology 4-balls? This is a well-known hard problem even for Seifert-fibred homology 3-spheres.

Is every slice knot a ribbon knot? How about for links? See 08/11/09 and 14/01/10.

Knots and Links

Can a knot type that has a primitive position in a genus two Heegaard surface for the 3-sphere also have a non-primitive position? Can a knot that does not have a primitive position have two different non-primitive positions? See 1/22/09.

Find knots in the 3-sphere such that the ratios between the grid number and the genus-one bridge number are arbitrarily high. See 11/25/08.

Is it possible to level an unknotting tunnel with respect to a level unknotted torus while keeping the knot in minimal bridge position with respect to the torus? See 1/24/08.

Are there sets of vertices in the width complex with bounded width but infinite diameter? See 1/15/08.

Is the width complex (with all higher dimensional cells) homotopy equivalent to Hatcher’s space of knots? See 1/15/08.

If an unknotting tunnel in a hyperbolic knot is isotopic to a geodesic in the knot complement, does it define a unique pair of meridians for the knot relative to the tunnel, and if so are these the same as those defined by leveling the tunnel with respect to a bridge sphere? See 1/11/08.

Is every knot in S^3 with a non-trivial lens space surgery a Berge knot? See 11/19/07 and 1/19/08.

Is every unknotting tunnel for a hyperbolic tunnel number one knot isotopic to a geodesic? See 11/18/07.

How is the minimal number of crossings in a knot or link diagram related to the JSJ-decomposition of the complement? Is crossing number additive under connect-sum? How does it behave for Whitehead doubles, and so on? See 25/03/12.

4-Manifolds

The smooth 4-dimensional Poincare conjecture. If a smooth 4-manifold has the homotopy-type of the 4-sphere, must it be diffeomorphic to the 4-sphere? By Freedman’s work, it must be homeomorphic to the 4-sphere.

Is there an algorithm to recognize a triangulated 4-sphere? See 07/11/09.

Misc.

Is there a Ricci-flow with surgery proof of the Smale Conjecture, that Diff(S^3) has the homotopy type of O(4)? See 10/8/03.

There are still several outstanding families of 3-manifolds for which the homotopy-type of the diffeomorphism group is not known, like RP^3. Also, for some the description is still rather complicated, for example for reducible 3-manifolds. Can one simplify these descriptions, and complete the descriptions for the unknown homotopy types, like Diff(RP^3)?

Does Diff(S^4) have the homotopy-type of O(5)?

If you would like to suggest an open question, feel free to leave a comment below. Since this is a permanent page, comments may be deleted after they have been addressed.

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Here is a problem due to Kropholler and Roller which is bothering me in retirement. It is possibly not very interesting for many but a special case of it turned out be useful and is one half of the algebraic torus theorem and was proved by Dunwoody and Roller in 1986. The question is this: Suppose G is group and H a subgroup, both finitely generated and assume that there is a non-trivial H-almost invariant set X with HXH=X. Then the ‘conjecture’ is that G splits over a subgroup commensurable with a subgroup of H. The algebraic hypothesis HXH=X can be reformulated in terms of strong crossings (done by Peter Scott and myself in the paper “Splittings of Groups and Intersection Numbers”). Apart from the result of Dunwoody Roller in the virtually polycyclic case (the proofs actually give more), there is a papaer of M.Sageev which proves the conjecture for quasiconvex subgroups of hyperbolic groups. I think that the result is true for 3-manifold groups and surface subgroups. It is a strange problem whose firther uses are not clear but it keeps bugging me.